Inside dynamics for stage-structured integrodifference equations

Nathan G Marculis (1) , Jimmy Garnier (2) , Roger Lui (3) , Mark A Lewis (1)

1 University of Alberta
2 LAMA - Laboratoire de Mathématiques
3 Worcester Polytechnic Institute

A stage-structured model of integrodifference equations is used to study the asymptotic neutral genetic structure of populations undergoing range expansion. That is, we study the inside dynamics of solutions to stage-structured integrodifference equations. To analyze the genetic consequences for long term population spread, we decompose the solution into neutral genetic components called neutral fractions. The inside dynamics are then given by the spatiotemporal evolution of these neutral fractions. We show that, under some mild assumptions on the dispersal kernels and population projection matrix, the spread is dominated by individuals at the leading edge of the expansion. This result is consistent with the founder effect. In the case where there are multiple neutral fractions at the leading edge we are able to explicitly calculate the asymptotic proportion of these fractions found in the long-term population spread. This formula is simple and depends only on the right and left eigenvectors of the population projection matrix evaluated at zero and the initial proportion of each neutral fraction at the leading edge of the range expansion. In the absence of a strong Allee effect, multiple neutral fractions can drive the long-term population spread, a situation not possible with the scalar model.

Mots clés: integrodifference equations; stage-structure; inside dynamics; neutral genetic structure; founder effect; founder effect Mathematics Subject Classification (2000) 39A10; 45G10; 92D25; 92D40

Domaines: Equations aux dérivées partielles [math.AP]; Ecologie, Environnement

Nathan G Marculis, Jimmy Garnier, Roger Lui, Mark A Lewis. Inside dynamics for stage-structured integrodifference equations. Journal of Mathematical Biology, 2019, ⟨10.1007/s00285-019-01378-9⟩. ⟨hal-02367984v2⟩